Moreover, from Section 3.2 in [1]}, the definition of the \(\dot{H}^{\alpha }\)  norm can be written as
\(\Vert u\Vert ^{2} _{\dot{H}^{\alpha }}= \int _{\mathbb {R}^{3}} |\xi |^{2\alpha }|\hat{u}(\xi )|^{2}d\xi =\int _{\mathbb {R}_{+}^{4}} y^{1-2\alpha } | \nabla ^{\ast } u^{\ast } |^{2}dxdy.\) 
